Fig. 58.
It consists of two beams, a b, 4' long, placed parallel to each other at a distance of 3"·5, and supported at each end; they are firmly clamped to the supports, and a roadway of short pieces is laid upon them. At the points of trisection of the beams c, d, struts c f and d e are clamped, their lower ends being supported by the framework: these struts are 2' long, and there are two of them supporting each of the beams. The tray g is attached by a chain to a stout piece of wood, which rests upon the roadway at the centre of the bridge.
423. We shall first determine the strength of this bridge by actual experiment, and then we shall endeavour to explain the results in accordance with mechanical principles. We can observe the deflection of the bridge by the cathetometer in the manner already described ([Art. 362]). By this means we shall ascertain whether the load has permanently injured the elasticity of the structure ([Art. 367]). We begin by testing the deflection when a load is distributed uniformly, as the weights are disposed in the case of [Fig. 62]. A cross is marked upon one of the beams, and is viewed in the cathetometer. We arrange 11 stone weights along the bridge, and the cathetometer shows that the deflection is only 0"·09: the elasticity of the bridge remains unaltered, for when the weights are removed the cross on the beam returns to its original position; hence the bridge is well able to bear this load.
424. We remove the row of weights from the bridge and suspend the tray from the roadway. I take my place at the cathetometer to note the deflection, while my assistant places weights h h on the tray. 1 cwt. being the load, I see that the deflection amounts to 0"·2; with 2 cwt. the deflection reaches 0·43"; and the bridge breaks with 238 lbs.
425. Let us endeavour to calculate the additional strength which the struts have imparted to the bridge. By [Table XXIV]. we see that a rod 40" × 0"·5 × 0"·5 is broken by a load of 19 lbs.: hence the beams of the bridge would have been broken by a load of 38 lbs. if their ends had been free. As, however, the ends of the beams had been clamped down, we learn from [Art. 411] that a double load would be necessary. We may, however, be confident that about 80 lbs. would have broken the unsupported bridge. The strength is, therefore, increased threefold by the struts, for a load of 238 lbs. was required to produce fracture.
426. We might have anticipated this result, because the points c and d being supported by the struts may be considered as almost fixed points; in fact, we see that c cannot descend, because the triangle a c f is unalterable, and for a similar reason d remains fixed: the beam breaks between c and d, and the force required must therefore be sufficient to break a beam supported at the points c and d, whose ends are secured. But c d is one-third of a b, and we have already seen that the strength of a beam is inversely as its length ([Art. 388]); hence the force required to break the beam when supported by the struts is three times as large as would have been necessary to break the unsupported beam. Thus the strength of the bridge is explained.
427. As a load of 238 lbs. applied near the centre is necessary to break this bridge, it follows from the principle of [Art. 408] that a load of about double this amount must be placed uniformly on the roadway before it succumbs; we can, therefore, understand how a load of 11 stone was easily borne (Art 423) without permanent injury to the elasticity of the structure. If we take the factor of safety as 3, we see that a bridge of the form we have been considering may carry, as its ordinary working load, a far greater weight than would have crushed it if unsupported by the struts and with free ends.
428. The strength of the bridge in [Fig. 58] is greater in some parts than in others. At the points c and d a maximum load could be borne; the weakest places on the bridge are in the middle points of the segments a c, d c, and d b. The load applied by the tray was principally borne at the middle of d c, but owing to the piece of wood which sustained the chain being about 18" long, the load was to some extent distributed.
The thrust upon the struts is not so easy to calculate accurately. That down c f for example must be less than if the part c d were removed, and half the load were suspended from c. The force in this case can be determined by principles already explained ([Art. 420]).